The aim of this thesis is twofold. First, we investigate spaces defined by asserting that their nowhere dense subsets have certain properties. Secondly, we develop some techniques for the construction of topological spaces. We consider spaces where the nowhere dense sets are asserted to have some property P, calling such spaces boundary-P. We show that if there are no Lusin spaces then every compact boundary-metrizable space is metrizable. Boundary-separability is also studied and we show that if there are no L-spaces then every boundary-separable space is separable. By adapting the absolute dimension function of Arhangel'skii, we define the new concept of cohesion. We show that every compact cohesive and every Hausdorff, sequential cohesive space is scattered. However, we construct regular, crowded spaces of all finite cohesions though there are no regular spaces of transfinite cohesion. We consider too the preservation of cohesion under various mappings and under the formation of products. Turning to construction, we consider the class of compact monotonically normal spaces. It is well-known that it contains the class of spaces which are the continuous images of compact ordered spaces but it is still open as to whether they are actually distinct classes. Using Watson's resolutions, we give a method for constructing monotonically normal spaces. Though this also preserves continuous images of arcs, we show that it is because of a powerful result of Cornette rather than any trivial observation. We also examine more closely monotone normality in images of compact ordered spaces using the Collins-Roscoe structuring mechanism. From this, we extract a strong instance of the mechanism, linear chain (F), which is held by all images of ordered compacta and all proto-metrizable spaces and implies Junnila's concept of utter normality. Elementary submodels are an important tool in the construction of topological spaces. We develop a general method for applying them in varying circumstances and illustrate it by constructing three examples: Balogh's Q-set space, Rudin's normal but not collectionwise Hausdorff space and Balogh's small Dowker space.

The purpose of this paper is to develop some properties of uniformly locally compact spaces. The terminology and symbology used are the same as those used in General Topology, by J. L. Kelley.

Doctor of Philosophy / Department of Mathematics / David Yetter / We generalize the definition of the framed Kontsevich integral initially presented in [LM1]. We study the behavior of the renormalized framed Kontsevich integral Z[hat]_f under band sum moves and show that it can be further renormalized into some invariant Z[widetilde]_f that is well-behaved under moves for which link components of interest are locally put on top of each other. Originally, Le, Murakami and Ohtsuki ([LM5], [LM6]) showed that another choice of normalization is better suited for moves for which link components involved in the band sum move are put side by side. We show the choice of renormalization leads to essentially the same invariant and that the use of one renormalization or the other is just a matter of preference depending on whether one decides to have a horizontal or a vertical band sum. Much of the work on Z[widetilde]_f relies on using the tangle chord diagrams version of Z[hat]_f ([ChDu]). This leads us to introducing a matrix representation of tangle chord diagrams, where each chord is represented by a matrix, and tangle chord diagrams of degree $m$ are represented by stacks of m matrices, one for each chord making up the diagram. We show matrix congruences for some appropriately chosen matrices implement on the modified Kontsevich integral Z[widetilde]_f the band sum move on links. We show how Z[widetilde]_f in matrix notation behaves under the Reidemeister moves and under orientation changes. We show that for a link L in plat position, Z_f(L) in book notation is enough to recover its expression in terms of chord diagrams. We elucidate the relation between Z[check]_f and Z[widetilde]_f and show the quotienting procedure to produce 3-manifold invariants from those as introduced in [LM5] is blind to the choice of normalization, and thus any choice of normalization leads to a 3-manifold invariant.

During the two last decades, topology optimization has grown to be an accepted and used method to produce conceptual designs. Topology optimization is traditionally carried out on a component level, but in this project, the possibility to apply it to airframe design on a full scale aeroplane model is evaluated. The project features a conceptual flying-wing design on which the study is to be carried out. Inertia Relief is used to constrain the aeroplane instead of traditional single point constraints with rigid body motion being suppressed by the application of accelerations instead of traditional forces and moments. The inertia relief method utilized the inertia of the aeroplane to achieve a state of quasi-equilibrium such that static finite element analysis can be carried out. Two load cases are used: a steep pitch-up manoeuvre and a landing scenario. Aerodynamic forces are calculated for the pitch-up load case via an in-house solver, with the pressure being mapped to the finite element mesh via a Matlab-script to account for different mesh sizes. Increased gravitational loads are used in the landing load case to simulate the dynamic loading caused in a real landing scenario, which is unable to be accounted for directly in the topology optimization. It can be concluded that the optimization is unable to account for one of the major design limitations: buckling of the outer skin. Approaches to account for the buckling of the outer skin are introduced and analysed, with a focus on local compression constraints throughout the wing. The compression constraints produce some promising results but are not without major drawbacks and complications. In general, a one-step topology optimization to produce a mature conceptual airframe design is not possible with optimization algorithms today. It may be possible to adopt a multiple-step optimization approach utilizing topology optimization with following size and shape optimization to achieve a design, which could be expanded on in a future project.

Nesse trabalho combinamos a teoria de Nielsen de pontos fixos com a teoria dos grupos de transformações para dar condições necessárias e sucientes para realizar um subconjunto A localmente contrátil de X G - como o conjunto de pontos xos de uma apli- h : X X em uma classe de homotopia equivariante dada, onde G é X é uma G -variedade suave e compacta. Além disso, se X é o espaço total de um G -brado localmente trivial demos condições cação equivariante um grupo de Lie compacto e necessárias e sucientes para o correspondente problema de realização para aplicações G -equivariantes que preservam bra, onde G é um grupo nito. / In this work, we combine the Nielsen fixed point theory with the transformation group theory to present necessary and sucient conditions for the realization of a locally contrac- G -subset A of X as the xed point set of a map h : X X in a given G -homotopy class. Here, G is a compact Lie group and X is a compact smooth G -manifold. In adition, if X is the total space of a G -ber bundle we present necessary and sucient conditions for the corresponding realization problem for G -ber-preserving maps when G tible is a nite group.

by Li Ho-chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 97-100). / Introduction --- p.iv / Chapter 1 --- Spectral spaces --- p.1 / Chapter 1.1 --- Basic notions --- p.1 / Chapter 1.1.1 --- Order compatible topology --- p.1 / Chapter 1.1.2 --- Prime spectrums and Zariski topology --- p.3 / Chapter 1.1.3 --- Lattice-ordered groups --- p.4 / Chapter 1.1.4 --- Spectral spaces and patch topology --- p.5 / Chapter 1.2 --- Properties of patches and the patch topology --- p.6 / Chapter 1.3 --- Properties of spectral spaces --- p.8 / Chapter 1.4 --- Another characterization of spectral spaces --- p.13 / Chapter 1.5 --- The maxspectral spaces --- p.14 / Chapter 2 --- The ordering on Spec(R) --- p.16 / Chapter 2.1 --- Two distinguished properties of a spectral poset --- p.17 / Chapter 2.2 --- Finite partially ordered sets --- p.18 / Chapter 2.3 --- Several classes of special rings --- p.19 / Chapter 2.4 --- Spectral trees --- p.23 / Chapter 2.5 --- Ordered disjoint unions --- p.26 / Chapter 2.6 --- Another necessary condition for a poset to be spectral --- p.31 / Chapter 2.7 --- Possible partial orderings for spectral posets --- p.33 / Chapter 3 --- The topology on Spec(R) --- p.35 / Chapter 3.1 --- Basic notions about Spec(R) --- p.35 / Chapter 3.2 --- The Zariski topology on Spec(R) --- p.37 / Chapter 3.2.1 --- Hausdorffness --- p.37 / Chapter 3.2.2 --- Irreducibility --- p.41 / Chapter 3.2.3 --- Connectedness --- p.45 / Chapter 3.2.4 --- Normality --- p.49 / Chapter 3.3 --- Topology on Min(R) and Baer rings --- p.55 / Chapter 4 --- Study algebraic properties from Spec(R) --- p.68 / Chapter 4.1 --- Prime spectrums of Bezout rings --- p.69 / Chapter 4.2 --- D-closed subsets of Spec(R) --- p.84 / Chapter 4.3 --- The C(m) topology --- p.88 / Chapter 4.4 --- Prime spectrum of Noetherian ring --- p.89 / Chapter 4.5 --- Reduced Bezout rings that are coherent --- p.93 / Chapter 4.6 --- Applications --- p.94 / Bibliography --- p.97

The Periodicity theorem of Hopkins and Smith tells us that any finite spectrum supports a $v_n$-map for some $n$. We are interested in finding finite $2$-local spectra that both support a $v_2$-map with a low power of $v_2$ and have few cells. Following the process outlined in Palmieri-Sadofsky, we study a related class of self-maps, known as $u_2$-maps, between stably finite spectra. We construct examples of spectra that might be expected to support $u_2^1$-maps, and then we use Margolis homology and homological algebra computations to show that they do not support $u_2^1$-maps. We also show that one example does not support a $u_2^2$-map. The nonexistence of $u_2$-maps on these spectra eliminates certain examples from consideration by this technique.

The topological complexity was introduced by Michael Farber in 2003 motivated by applications of algebraic topology to robotics. It is a numerical homotopy invariant of a space which measures the instability of motion planning. In this thesis we determine this invariant for unordered configuration spaces of surfaces in many cases and reduce it to a few possible values in other cases. We also determine the topological complexity of mixed configuration spaces and related spaces. In contrast to the ordered configuration spaces, these computations remained elusive because the standard methods do not work here, as we argue in the Appendix. Apart from the interest from the motion planning perspective to decide whether unordered or ordered configurations have a higher topological complexity, there is another motivation. Namely, it is an open problem to give an algebraic description of the topological complexity of an aspherical space in terms of the fundamental group. The spaces under consideration are aspherical and so the topological complexity (being a homotopy invariant) becomes an invariant of their fundamental groups, the surface braid groups. The computation of the topological complexity of surface braid groups and their finite index subgroups thus provides further examples which might help tackle this open problem. Furthermore, the results could be used to gain information about the subgroup structure of surface braid groups. Often the topological complexity is calculated indirectly without actually finding an optimal motion planner which realizes it. Nonetheless, in some cases we will construct explicit motion planners and then prove that they are optimal. All those motion planners are collected in the last chapter.

We propose a new type of state sum model for two-dimensional surfaces that takes into account topology and spin. The definition used - new to the literature - provides a rich class of extended models called spin models. Both examples and general properties are studied. Most prominently, we find this type of model can depend on a surface spin structure through parity alone and we explore explicit cases that feature this behaviour. Further directions for the two dimensional world are analysed: we introduce a source of new information - defects - and show how they can enlarge the class of spin models available.

Dehn surgery and the notion of reducible manifolds are both important tools in the study of 3-manifolds. The Cabling Conjecture of Francisco González-Acuña and Hamish Short describes the purported circumstances under which Dehn surgery can produce a reducible manifold. This thesis extends the work of James Allen Hoffman, who proved the Cabling Conjecture for knots of bridge number up to four. Hoffman built upon the combinatorial machinery used by Cameron Gordon and John Luecke in their solution to the knot complement problem. The combinatorial approach starts with the graphs of intersection of a thin level sphere of the knot and the reducing sphere in the surgered manifold. Gordon and Luecke's proof then proceeds by induction on certain cycles. Hoffman provides more insight into the structure of the base case of the induction (i.e. in an innermost cycle or a graph containing no such cycles). Hoffman uses this structure in a case-by-case proof of the Cabling Conjecture for knots of bridge number up to four. We find trees with specific properties in the graph of intersection, and use them to provethe existence of structure which provides lower bounds on the number of the aforementioned innermost cycles. Our results combined with a recent lower bound on the number of vertices inside the innermost cycles succinctly prove the conjecture for bridge number up to five and suggests an approach to the conjecture for knots of higher bridge number.